Let \(f\in E^2(p,w)\) . For every \(h\in E^2(p,w)\) , since \(C(p,w)<\infty \) , we can use Theorem REF  to obtain
\(\mathbb {E}_{p}[h\widetilde{\mathcal {L}}f\,w]=-\mathbb {E}_p\left[h\widetilde{\mathcal {T}}If\right]=\mathbb {E}_p\left[Ih\,If\right].\) 
Moreover, we have
\(\mathbb {E}_p\left[Ih\,If\right]\le \left\Vert Ih\right\Vert _{L^2(p)}\left\Vert If\right\Vert _{L^2(p)} \le C(p,w) \left\Vert f\right\Vert _{L^2(pw)}\left\Vert h\right\Vert _{L^2(pw)}.\) 
Putting these two computations together, we see that the form \(h\in E^2(p,w)\mapsto \mathbb {E}_{p}[h\widetilde{\mathcal {L}}f\,w]\)  is linear and continuous. By the Riesz representation theorem, there exists a \(g\in E^2(p,w)\)  such that \(\mathbb {E}_{p}[h\widetilde{\mathcal {L}}f\,w]=\mathbb {E}_{p}[h g\,w]\)  for all \(h\in E^2(p,w)\) . As \(pw\in L^1_{\text{loc}}(]a,b[)\) , \(E^2(p,w)\)  contains the indicator functions of compact sets.
So, we have
\(\int _E (\widetilde{\mathcal {L}}f-g)pw=0\) 
for all compact set \(E\subset ]a,b[\) . This implies that \(\widetilde{\mathcal {L}}f=g\)  a.e. We conclude that \(\widetilde{\mathcal {L}}f\in E^2(p,w)\)  and \(\widetilde{\mathcal {L}}\)  is well defined. Equation (REF ) also shows that \(\widetilde{\mathcal {L}}\)  is self-adjoint and positive, in the sense that \(\mathbb {E}_{p}[f\widetilde{\mathcal {L}}f\,w]\ge 0\)  for all \(f\in E^2(p,w)\) .
The Hellinger-Toeplitz Theorem says that a self-adjoint operator defined everywhere on a space is continuous on this space (see the Corollary of Theorem III.12 in [1]}). Hence, \(\widetilde{\mathcal {L}}\)  is continuous. Another way of seeing that \(\widetilde{\mathcal {L}}\)  is continuous is to take \(h=\widetilde{\mathcal {L}}f\)  in (REF ). Finally, as \(\widetilde{\mathcal {L}}\)  is self-adjoint, we have by Proposition 2.13 in Chapter 2 of [2]}
\(\left\Vert \widetilde{\mathcal {L}}\right\Vert _{E^2(p,w)\rightarrow E^2(p,w)}=\sup _{f\in E^2(p,w),\left\Vert f\right\Vert =1}\mathbb {E}_{p}\left[f\widetilde{\mathcal {L}}fw\right]=\sup _{f\in E^2(p,w),\left\Vert f\right\Vert =1}\mathrm {Var}_p\left[If\right].\) 
Moreover, using that \(I\)  is a bijection, we can see that
\(\sup _{f\in E^2(p,w),\left\Vert f\right\Vert _{E^2}=1}\mathrm {Var}_p\left[If\right]=\sup _{h\in H^1_c(p,w),\left\Vert h^{\prime }\right\Vert _{E^2}=1}\mathrm {Var}_p\left[h\right]=C(p,w).\) 
With these two computations, we get the desired conclusion.
